Bad Notation? Using the infinity symbol with inequality signs. I am a teacher and I teach my students that using the infinity symbol as pictured here is bad.  Students learn set notation along side interval notation, and when translating directly between the two, often times what should be written $x<2$ becomes $-\infty<x<2$. Now I have discovered that the textbook my county adopted uses the notation the way I instruct my students NOT to, and I am looking for confirmation that I have not been misleading my students. Please take a side - I say NEVER put an infinity sign in an inequality expression!  Wrong or right?  Gray area?  TIA. The following is from the textbook:

What is the domain of $x=-9y^2$?
A.  $-\infty<x\leq-3$
B.  $-\infty<x\leq0$
C.  $-\infty<x<\infty$
D.  $0\leq x<\infty$

 A: There is no "right" or "wrong" here. This is a matter of convention and I think either convention is okay. The comments make good points that this is not so different from using $\infty$ in interval notation such as $[0, \infty)$ (which is fairly standard as far as I know) and also that it's consistent with the common notation $\sum a_i < \infty$ used to mean that a series converges (which is also fairly common as far as I know, and is used e.g. to specify that functions have bounded norm and stuff like that, in expressions like $\int |f(x)|^2 \, dx < \infty$).
However, for this specific question about domains I personally think this notation has the potential to be confusing for students and should be avoided. The reason is that writing a strict inequality $-\infty < x < \infty$ suggests the question: why not allow a weak inequality $-\infty \le x \le \infty$ into the notation? And then: if we allow $x$ to take the values $\pm \infty$ then why not allow $y$ to take these values? And then: should we say that e.g. $0$ is in the domain of $y = \frac{1}{x}$ because $y$ can take the value $\infty$ there? In other words, as Dietrich says in the comments, this notation invites confusion about whether $\infty$ is being treated as a real number, which makes the whole concept of domain ambiguous.
The interval notation $[0, \infty)$ also invites this confusion, admittedly, but it's very convenient and meanwhile in this context, as you say, the $\infty$s can be dropped entirely. The options could just be rewritten

*

*$x \le -3$

*$x \le 0$

*All $x$

*$0 \le x$
